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GT CalVa R MCCANN
Lundi 9 janvier 2017
Robert MC CANN (University of Toronto Bahen Centre)
Free Discontinuities in Optimal Transport
Abstract : Optimal maps in $R^n$ to disconnected targets necessarily contain discontinuities (i.e. tears). But how smooth are these tears ? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions --- DC for short. If in addition the targets are convex the tears are actually $C^1,\alpha$. Similarly, under suitable affine independence assumptions, singularities of multiplicity $k$ lie on DC rectifiable submanifolds of dimension $n+1-k$. These are stable with respect to $W_\infty$ perturbations of the target measure. Moreover, there is at most one singularity of multiplicity $n$. This represents joint work with Jun Kitagawa.